On the largest prime factors less than y of consecutive shifted primes

Abstract

For an integer n > 1, let P⁺(n) be the largest prime factor of n, and let P_y⁺(n) denote the largest prime factor of n not exceeding y. One of Erdős and Turán's conjectures asserts that the asymptotic density of integers n satisfying P⁺(n)<P⁺(n+1) is 1/2. Furthermore, Wang conjectures that for x→∞, one has #{p≤ x:P⁺(p−1)<P⁺(p+1)}∼ π(x)/2. In this paper, we show the following results. For any 3≤ y≪ x^o(1), for x→∞, we have &#{n≤ x:P⁺_y(n)<P⁺_y(n+1)}∼ 1/2x, &#{p≤ x:P⁺_y(p−1)<P⁺_y(p+1)}∼ 1/2π(x), where the first result extends a previous range 3≤ y≪ exp(log x/(100loglog x)) due to Rivat (2001). For any 0<α<17/32, let y=x^α. Then there exists h(α)>0 such that #{p≤ x:P_y⁺(p−1)<P_y⁺(p+1)}≥(h(α)+o(1))π(x). In particular, the function h satisfies lim_{α→ 0⁺}h(α)=1/2. Similar result also holds for #{n≤ x:P_y⁺(n)<P_y⁺(n+1)}, which improves a previous result lim_{α→ 0⁺} h(α)=1/4 due to Wang (2019).

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BibTeX RIS

BibTeX

@misc{yang2026largest,
  title = {{On the largest prime factors less than $y$ of consecutive shifted primes}},
  author = {Zhiyuan Yang},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2609.15166},
  eprint = {2609.15166},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.15166},
}

RIS

TY  - UNPB
TI  - On the largest prime factors less than y of consecutive shifted primes
AU  - Zhiyuan Yang
PY  - 2026
DA  - 2026/10/29
PB  - arXiv
DO  - 10.48550/arxiv.2609.15166
UR  - https://arxiv.org/abs/2609.15166
AB  - For an integer n > 1, let P^+(n) be the largest prime factor of n, and let P_y^+(n) denote the largest prime factor of n not exceeding y. One of Erdős and Turán's conjectures asserts that the asymptotic density of integers n satisfying P^+(n)<P^+(n+1) is 1/2. Furthermore, Wang conjectures that for x→∞, one has #{p≤ x:P^+(p−1)<P^+(p+1)}∼ π(x)/2. In this paper, we show the following results. For any 3≤ y≪ x^o(1), for x→∞, we have &#{n≤ x:P^+_y(n)<P^+_y(n+1)}∼ 1/2x, &#{p≤ x:P^+_y(p−1)<P^+_y(p+1)}∼ 1/2π(x), where the first result extends a previous range 3≤ y≪ exp(log x/(100loglog x)) due to Rivat (2001). For any 0<α<17/32, let y=x^α. Then there exists h(α)>0 such that #{p≤ x:P_y^+(p−1)<P_y^+(p+1)}≥(h(α)+o(1))π(x). In particular, the function h satisfies lim_α→ 0^+h(α)=1/2. Similar result also holds for #{n≤ x:P_y^+(n)<P_y^+(n+1)}, which improves a previous result lim_α→ 0^+ h(α)=1/4 due to Wang (2019).
ER  -

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